View source for Boolean-valued model

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A model defined as follows. Let  $  \Omega $
be the signature of some first-order language  $  L $
with one kind of variables, i.e.  $  \Omega $
is the set of symbols of functions and predicates. A Boolean-valued model then is a triple  $  M = (B _ {M} , V _ {M} , \Omega _ {M} ) $, 
where  $  B _ {M} $
is a non-degenerate [[Boolean algebra|Boolean algebra]],  $  V _ {M} $
is a non-empty set, and  $  \Omega _ {M} $
is a function defined on  $  \Omega $
such that

$$ 
\Omega _ {M} ( \rho )  \in \ 
V _ {M} ^ {V _ {M}  ^ {n} }
$$

if  $  \rho $
is an  $  n $-
place function symbol, and

$$ 
\Omega _ {M} ( \rho )  \in \ 
B _ {M} ^ {V _ {M}  ^ {n} }
$$

if  $  \rho $
is an  $  n $-
place predicate symbol. The symbol  $  X  ^ {Y} $
denotes the set of all functions defined on  $  Y $
with values in  $  X $
and  $  X  ^ {n} = X ^ {\{ {m } : {m<n } \} } $, 
where  $  n \geq  0 $
is a natural number. The Boolean algebra  $  B _ {M} $
is called the set of truth values of the model  $  M $. 
The set  $  V _ {M} $
is called the universe of  $  M $. 
A Boolean-valued model  $  M $
is also called a  $  B $-
model if the set of truth values is the Boolean algebra  $  B $, 
$  B _ {M} = B $. 
If a Boolean algebra  $  B $
is a two-element algebra (i.e.  $  B = \{ 0, 1 \} $), 
then the  $  B $-
model  $  M $
is the classical two-valued model.

Let  $  L _ {M} $
be a language, complemented by new individual constants: each  $  v \in V _ {M} $
having its own individual constant  $  \mathbf v $. 
Let  $  M $
be a  $  B $-
model and let  $  B = (B;  0, 1, C, \cup , \cap ) $
be a complete Boolean algebra; the equalities 1)–8) below then define the value  $  \| e \| _ {M} $
of each closed expression  $  e $(
i.e. of a formula or a term without free variables) of  $  L _ {M} $:

1)  $  \| \mathbf v \| _ {M} = v $, 
where  $  v \in V _ {M} ; $

2)  $  \| \rho ( \tau _ {1} \dots t _ {n} ) \| _ {M} = ( \Omega _ {M} ( \rho )) ( \| \tau _ {1} \| _ {M} \dots \| \tau _ {n} \| _ {M} ), $
where  $  \tau _ {1} \dots \tau _ {n} $
are closed terms and  $  \rho $
is an  $  n $-
place function or predicate symbol;

3)  $  \| \phi \supset \psi \| _ {M} = - \| \phi \| _ {M} \cup \| \psi \| _ {M} ; $

4)  $  \| \phi \lor \psi \| _ {M} = \| \phi \| _ {M} \cup \| \psi \| _ {M} ; $

5)  $  \| \phi \wedge \psi \| _ {M} = \| \phi \| _ {M} \cap \| \psi \| _ {M} ; $

6)  $  \| \neg \phi \| _ {M} = - \| \phi \| _ {M} ; $

7)  $  \| \exists \xi  \phi ( \xi ) \| _ {M} = \cup _ {v \in V _ {M}  } \| \phi ( \mathbf v ) \| _ {M} ; $

8)  $  \| \forall \xi  \phi ( \xi ) \| _ {M} = \cap _ {v \in V _ {M}  } \| \phi ( \mathbf v ) \| _ {M} . $

The relations 1)–8) define the value  $  \| e \| _ {M} $
for certain non-complete Boolean algebras as well; the only condition is that the infinite unions and intersections in 7) and 8) exist. The concept of a Boolean-valued model can also be introduced for languages with more than one kind of variables. In such a case each kind of variable has its own domain of variation  $  V _ {M} $.

A closed formula  $  \phi $
is said to be true in a  $  B $-
model  $  M $(
$  M \vDash \phi $) 
if

$$ 
\| \phi \| _ {M}  =  1.
$$

A  $  B $-
model  $  M $
is said to be a model of a theory  $  T $
if  $  M \vDash \phi $
for all axioms  $  \phi $
of  $  T $. 
If  $  h $
is a homomorphism of a Boolean algebra  $  B $
into a Boolean algebra  $  B ^ { \prime } $
preserving infinite unions and intersections, then there exists a  $  B ^ { \prime } $
model  $  M  ^  \prime  $
such that

$$ 
\| \phi \| _ {M  ^  \prime   }  = \ 
h ( \| \phi \| _ {M} )
$$

for each closed formula  $  \phi $
of  $  L _ {M} $. 
If the universe of a model  $  M $
is countable, then there exists a homomorphism  $  h $
into the Boolean algebra  $  \{ 0, 1 \} $, 
under which  $  M $
is transformed into the classical two-valued model  $  M  ^  \prime  $
such that

$$ 
M \vDash \phi  \rightarrow  M  ^  \prime  \vDash \phi .
$$

It has been shown that a theory  $  T $
is consistent if and only if it has a Boolean-valued model. This theorem forms the basis of the application of the theory of Boolean-valued models to problems of the consistency of axiomatic theories.

If the Boolean-valued model of a theory  $  T $
is constructed by means of another axiomatic theory  $  S $, 
then one obtains the statement on the consistency of  $  T $
relative to  $  S $. 
Thus, the result due to P. Cohen on the consistency of the theory  $   \mathop{\rm ZF} + (2 ^ {\aleph _ {0} } > \aleph _ {1)} $
relative to ZF is obtained by constructing the respective Boolean-valued model by means of the system ZF (cf. [[Forcing method|Forcing method]]). The construction of the Cohen forcing relation  $  p \Vdash \phi $
is equivalent to that of a Boolean-valued model  $  M $
such that

$$ 
\| \phi \| _ {M}  =  \{ {p } : {p \Vdash \neg \neg \phi } \}
.
$$

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Rasiowa,   R. Sikorski,   "The mathematics of metamathematics" , Polska Akad. Nauk  (1963)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  T.J. Jech,   "Lectures in set theory: with particular emphasis on the method of forcing" , ''Lect. notes in math.'' , '''217''' , Springer  (1971)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  G. Takeuti,   W.M. Zaring,   "Axiomatic set theory" , Springer  (1973)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.I. Manin,   "The problem of the continuum"  ''J. Soviet Math.'' , '''5''' :  4  (1976)  pp. 451–502  ''Itogi Nauk. i Tekhn. Sovrem. Problemy'' , '''5'''  (1975)  pp. 5–73 {{ZBL|0375.02055}}</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Bell,   "Boolean-valued models and independence proofs in set theory" , Clarendon Press  (1977)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Jech,   "Set theory" , Acad. Press  (1978)  (Translated from German)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen,   "Set theory" , North-Holland  (1980)</TD></TR>
</table>